P
P
-G
E
M
M A S T
ESTUDO DA FORMAÇÃO DE GEADA EM TROCADORES
DE CALOR DE ALETAS PERIFÉRICAS
Florianópolis, SC
ESTUDO DA FORMAÇÃO DE GEADA EM TROCADORES
DE CALOR DE ALETAS PERIFÉRICAS
Dissertação submetida ao Programa de Pós-Graduação da Universidade Federal de Santa Catarina para obtenção do título de Mestre em Engenharia Mecânica. Orientador: Prof. Jader Riso Barbosa Jr., Ph.D.
Florianópolis, SC
Timmermann, Marco Aurélio Stimamiglio
Estudo da Formação de Geada em Trocadores de Calor de Aletas Periféricas / Marco Aurélio Stimamiglio
Timmermann; orientador, Jader Riso Barbosa Jr. – Florianópolis, SC, 2016.
200 p.
Dissertação (mestrado) – Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia Mecânica.
Inclui referências
1. Engenharia Mecânica. 2. Trocadores de Calor. 3. Aletas Periféricas. 4. Análise Experimental. 5. Modelagem Matemática. I. Barbosa Jr., Jader Riso. II. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia Mecânica. III. Título
ESTUDO DA FORMAÇÃO DE GEADA EM TROCADORES
DE CALOR DE ALETAS PERIFÉRICAS
Esta dissertação foi julgada adequada para a obtenção do título de “Mestre em Engenharia Mecânica” e aprovada em sua forma final pelo Programa de
Pós-Graduação em Engenharia Mecânica.
Florianópolis, 13 de dezembro de 2016
Prof. Jader Riso Barbosa Jr., Ph.D. - Orientador
Prof. Jonny Carlos da Silva, Dr.Eng. - Coordenador do Curso
BANCA EXAMINADORA
Prof. Alexandre Kupka da Silva, Ph.D.
Prof. Christian J. L. Hermes, Dr.Eng. (UFPR)
Primeiramente, gostaria de agradecer ao professor Jader Riso Barbosa Jr., por ter oferecido a oportunidade de realizar este trabalho, pela disponibilidade para discussões, pela dedicação e comprometimento com o projeto e pelos anos de ensino que tive sob sua orientação.
Aos meus familiares: meu pai Mário, minha mãe Susana e minha irmã Suzi, por me darem todo o suporte que preciso, por todo o carinho e amor.
À minha namorada Graziela, pelo amor, companheirismo e suporte nas etapas finais do trabalho.
Aos companheiros de laboratório Renata Rametta, Carla Rametta, Johann Barcelos, Joel Boeng e Fernando Knabben, pela amizade, discussões, chur-rascos e viagens, e em especial ao Rodolfo Espíndola, pelos anos de amizade e companheirismo dentro e fora do POLO, e pelas frequentes discussões e contribuições para este trabalho.
Aos colegas do CO2 Guilherme Zanotelli, Diego Marchi e Igor Galvão,
pelos chimarrões e pelas valiosas contribuições.
Aos colegas do POLO Bruno Yuji, Alan Nakashima, Tiago Melo, Jaime Lozano, Gustavo Coelho e Marco Diniz, pelo companheirismo e pelas con-truibuições.
Aos colegas de laboratório Pedro Cardoso, Rafael Lima, Alexsandro Sil-veira, Jorge Lubas, Jean Backer, Willian Longen, Marcelo Ribeiro, Amarilho Kruger e Deivid de Oliveira, cujas contribuições deram o suporte necessário para a conclusão deste trabalho.
Ao Fábio, Samuel, Jéssica, Luisa e Gabriela, pela amizade, companhia e bons momentos, e por estarem presentes em boa parte do desenvolvimento deste trabalho.
À Gabriela, Dalvani, Sérgio, Leondina, Vinicius e Luiza, pelas incontáveis vezes que me acolheram em suas casas e pelos bons momentos.
Ao POLO e seus demais colaboradores, por disponibilizar o espaço, ma-teriais, serviços e equipamentos necessários.
Aos membros da banca examinadora, por se disporem a avaliar e pelas contribuições ao trabalho.
À todos os amigos e familiares que não foram citados, mas que certamente contribuíram para o resultado deste trabalho.
nature, and to learn how things work gives you power to influence events, gives you power to help people who may need it, power to help yourselves, to shape a trajec-tory. So when I think of, “What is the meaning of life?”, to me that is not an eternal, unanswerable question. To me, that is in arms reach of me every day.”
Este trabalho apresenta um estudo teórico e experimental da formação de geada em trocadores de calor de aletas periféricas, os quais possuem uma nova geometria da superfície externa de trocadores de calor compactos. Tal estudo tende a complementar o trabalho desenvolvido por Pussoli (2010), o qual avaliou a performance termo-hidráulica deste tipo de trocador de calor sob condições secas (sem formação de condensado ou geada). A geometria ex-terna consiste em seis aletas cujas bases são acopladas radialmente ao redor do tubo, e cujas pontas são conectadas pelas aletas periféricas. Isto resulta em um conjunto hexagonal de aletas, que são fabricados em três diferentes tamanhos, caracterizados pelo comprimento das aletas radiais. As aletas são montadas sobre o tubo, defasadas de 30° entre si, dando origem ao trocador de calor. O trabalho foi dividido em duas frentes. A experimental consistiu em utilizar um calorímetro de túnel de vento de circuito fechado para testar um protótipo do trocador de calor de aletas periféricas, sob condições nas quais ocorre a formação de geada. O calorímetro controla a vazão volumétrica e as condições psicrométricas do ar na entrada da seção de teste, além da tem-peratura de entrada do fluido refrigerante que circula no trocador de calor. A taxa de transferência de calor e a massa de geada acumulada durante os testes são calculados por meio de balanços de energia e massa no trocador. Além disso, mede-se a queda de pressão no lado do ar causado pela presença da geada. A frente numérica consistiu em desenvolver a modelagem matemática dos fenômenos de transferência de calor e de massa que ocorrem no trocador para uma variedade de condições de operação, de modo a prever seu com-portamento transiente. O modelo considera o caminho percorrido pelo ar no trocador de calor como um meio poroso, cuja porosidade, diâmetro equiva-lente de partícula e propriedades termofísícas variam com o tempo devido ao acúmulo de geada. O equacionamento foi implementado na plataforma EES (Klein e Alvarado, 2015) para automatização dos cálculos. Os resultados ger-ados pelo modelo matemático apresentaram concordância satisfatória com os experimentos para as principais variáveis avaliadas. O modelo foi capaz de prever a redução na taxa de transferência de calor e o aumento da queda de
apresentou boa concordância no início de todas as simulações, com desvios da ordem de±15 %. Contudo, com o aumento do tempo de simulação, erros maiores foram encontrados em algumas condições. A comparação da massa de geada formada e da queda de pressão do lado do ar foram satisfatórias, tendo em vista que a maior parte das simulações correlacionou os dados ex-perimentais com desvios de±20 %.
Palavras-chave: trocador de calor, aletas periféricas, formação de geada, es-tudo experimental, modelagem matemática.
This work presents a theoretical and experimental study of frost formation in peripheral finned-tube (PFT) heat exchangers, which have a novel air-side geometry of compact heat exchangers. This study should complement the work developed by Pussoli (2010), who evaluated the thermo-hydraulic per-formance of these heat exchangers under dry conditions (with no frost or condensate formation). The air-side geometry consists in six fins radially dis-posed around the tube, connected on its bases to the tube and on its tips by the peripheral fins. This results in an hexagonal set of fins, which are fabricated in three sizes, each characterized by the length of the radial fin. The fins are mounted around the tube with a 30° offset from its neighbors. This work was divided in two parts. The experimental part consisted in using a closed-loop wind tunnel calorimeter to test a prototype of the peripheral finned-tube heat exchanger under frosting conditions. The apparatus controls the air flow rate and psychrometric conditions at the test section inlet and the coolant temper-ature at the heat exchanger inlet. The heat transfer rate and the mass of frost accumulated during the tests are calculated using the energy and mass bal-ances on the heat exchanger. Also, the change of the air-side pressure drop caused by frost is measured. The theoretical part consisted in developing a mathematical model of the heat and mass transfer phenomena that take place on the heat exchanger for a variety of test conditions, aiming to predict their transient behavior. The model treats the air flow paths as a porous medium, whose porosity, mean particle diameter and thermal properties vary with time due to the frost accumulation. The equations were implemented in the EES platform (Klein and Alvarado, 2015) to automate the calculation. The results obtained by the mathematical model presented satisfactory agreement with the experimental data for the main variables evaluated. The model was able to predict the reduction on the heat transfer rate and the increase on the air-side pressure drop due to the presence of frost. The enthalpy effectiveness pre-sented a good agreement at the early times of all simulation, with errors within
±15 % bands. However, as time increased in the simulations, larger errors were obtained in some of the conditions. The comparison of the mass of frost
agreement, as most of the results were correlated with the experimental data within±20 % error bands.
Keywords: heat exchanger, peripheral finned-tube, frost formation, experi-mental study, mathematical modeling.
Figure 1.1 – Harvesting of ice was a strong economic activity in the 1800’s . . . 29 Figure 1.2 – First refrigeration systems . . . 30 Figure 1.3 – Standard vapor compression refrigeration cycle . . . 31 Figure 1.4 – P-h diagram of the standard vapor compression
refrigera-tion cycle . . . 32 Figure 1.5 – Frost formation over an evaporator surface . . . 34 Figure 1.6 – (1) Individually finned tubes; (2) flat (continuous) fins on
an array of tubes (Shah and Sekulic, 2003) . . . 35 Figure 1.7 – (a) plain fins; (b) wavy fins; (c) louvered fins (Mao et al.,
2013) . . . 36 Figure 1.8 – Peripheral finned-tube geometry . . . 37 Figure 1.9 – Representation of the blockage caused by frost in the PFT
geometry. Adapted from Wu et al. (2007) . . . 37 Figure 1.10–Air flow across the peripheral fins (Wu et al., 2007) . . . 38 Figure 1.11–Thermal conductance per unit volume as a function of the
pumping power per unit volume for five types of air-side enhanced surfaces . . . 39 Figure 2.1 – Water phase diagram . . . 43 Figure 2.2 – Ice crystal shapes as a function of local conditions
(Lib-brecht, 2005) . . . 47 Figure 3.1 – Schematic cooling and dehumidifying of moist air. Adapted
from Threlkeld et al. (1998) . . . 66 Figure 3.2 – Schematic illustration of a fin covered with a layer of frost.
Adapted from Threlkeld et al. (1998) . . . . 68 Figure 3.3 – Schematic illustration of a tube-fin heat exchanger Adapted
from Threlkeld et al. (1998) . . . 71 Figure 4.1 – Photograph of the closed-loop wind tunnel calorimeter . . 79 Figure 4.2 – Schematic diagram of the wind tunnel . . . 80 Figure 4.3 – Schematic diagram of the test section (Silva, 2012) . . . 81 Figure 4.4 – Heat exchanger installed in the test section . . . 82
Figure 4.6 – Model of the tunnel, with the main components of the air conditioning system indicated by numbers . . . 85 Figure 4.7 – Heat exchanger after being removed from the test section 87 Figure 4.8 – Geometric characteristics of an hexagonal set of fins . . . 88 Figure 4.9 – Geometric characteristics that define different PFT heat
exchangers . . . 89 Figure 4.10–PFT heat exchanger prototype used in the experiments . . 90 Figure 4.11–Schematic diagram of the heat exchanger, and the main
variables involved . . . 91 Figure 4.12–Comparison of the heat transfer rates on the air and coolant
side for the entire data set . . . 94 Figure 4.13–Outlet air enthalpy variation with time on test 3 . . . 95 Figure 4.14–Outlet coolant temperature variation with time on test 3 . 95 Figure 4.15–Photographs of three different moments during test 8 . . . 96 Figure 5.1 – Illustration of a T-unit of a fin arrangement covered in frost 98 Figure 5.2 – Fin dimensions . . . 99 Figure 5.3 – Division of the heat exchanger in control volumes . . . . 103 Figure 5.4 – Frost layer enthalpy profile . . . 108 Figure 5.5 – Schematic representation of the contraction, expansion and
core pressure drop of a heat exchanger . . . 111 Figure 5.6 – Contraction and expansion coefficients as a function of the
solid matrix porosity . . . 113 Figure 5.7 – Division of the heat exchanger in control volumes, and the
main variables involved . . . 116 Figure 5.8 – User interface of the program . . . 118 Figure 5.9 – Flow chart of the solution algorithm . . . 119 Figure 6.1 – Comparison of the mass balance measurements and the
mass of frost weighed on a scale . . . 122 Figure 6.2 – Total mass of frost collected after each test . . . 123 Figure 6.3 – Uneven frost accumulation on the first row of tubes. The
arrows indicate the coolant flow direction . . . 124 Figure 6.4 – Typical irregular frost formation on a fin . . . 125
dicates a R3 fin and the blue arrow a R2 fin . . . 125
Figure 6.6 – Mass of frost variation with time, test 3 . . . 126
Figure 6.7 – Mass of frost variation with time, test 5 . . . 127
Figure 6.8 – Mass of frost variation with time, test 15 . . . 127
Figure 6.9 – Comparison of the frost accumulation predictions with the experimental data . . . 128
Figure 6.10–Average frost density variation with time, for all CVs, test 3129 Figure 6.11–Average frost density variation with time, for all CVs, test 15 . . . 130
Figure 6.12–Frost density variation with time, in CV 1, test 3 . . . 131
Figure 6.13–Frost density variation with time, in CV 5, test 3 . . . 131
Figure 6.14–Frost density variation with time, in CV 1, test 15 . . . . 132
Figure 6.15–Frost density variation with time, in CV 5, test 15 . . . . 132
Figure 6.16–Average frost thickness variation with time for all CVs, test 3 . . . 133
Figure 6.17–Average frost thickness variation with time for all CVs, test 15 . . . 134
Figure 6.18–Frost thickness variation with time, in CV 1, test 3 . . . . 134
Figure 6.19–Frost thickness variation with time, in CV 5, test 3 . . . . 135
Figure 6.20–Frost thickness variation with time, in CV 1, test 15 . . . 135
Figure 6.21–Frost thickness variation with time, in CV 5, test 15 . . . 136
Figure 6.22–Fin average temperature variation with time, in CV 5, test 3 137 Figure 6.23–Frost surface temperature variation with time, in CV 5, test 3137 Figure 6.24–Frost surface area variation with time for all CVs, test 3 . 138 Figure 6.25–Frost surface area variation with time for all CVs, test 15 139 Figure 6.26–Frost surface area variation with time for all fin types, in CV 1, test 3 . . . 140
Figure 6.27–Pressure drop variation with time, test 3 . . . 141
Figure 6.28–Pressure drop variation with time, test 5 . . . 142
Figure 6.29–Pressure drop variation with time, test 15 . . . 142 Figure 6.30–Variation of frost thickness and porosity for all CVs, test 3 143
drop results for the entire data set . . . 144
Figure 6.32–Heat exchanger pressure drop as a function of the mass of frost accumulated, for some of the test conditions . . . . 145
Figure 6.33–Heat exchanger pressure drop as a function of the average frost thickness in all CVs, for some of the test conditions 146 Figure 6.34–Heat transfer rate variation with time, test 3 . . . 147
Figure 6.35–Heat transfer rate variation with time, test 5 . . . 148
Figure 6.36–Heat transfer rate variation with time, test 15 . . . 148
Figure 6.37–Heat transfer rate variation with time for all fin types, in CV 1, test 3 . . . 149
Figure 6.38–Comparison of the experimental and numerical heat trans-fer rate results for the entire data set . . . 150
Figure 6.39–Comparison of the experimental and numerical enthalpy effectiveness results for the entire data set . . . 151
Figure 6.40–Change of the flow path due to frost thickening. Adapted from Wu et al. (2007) . . . 152
Figure 6.41–Frost average thermal conductivity variation with time, test 3 . . . 153
Figure 6.42–Frost average thermal conductivity variation with time, test 15 . . . 154
Figure 6.43–Heat exchanger thermal resistances variation with time, test 3 . . . 155
Figure 6.44–Heat exchanger thermal resistances variation with time, test 15 . . . 155
Figure A.1 – Dimensions used to calculate the geometric parameters of the fins . . . 174
Figure A.2 – Surface area of a radial fin . . . 175
Figure A.3 – Surface area of a peripheral fin . . . 176
Figure A.4 – Solid area of an hexagonal set of fins . . . 177
Figure A.5 – Exposed area of a fin . . . 178
Figure A.6 – Demonstration of the areas associated to the upper fillet surface and the fillet arc surface . . . 179
Figure B.1 – Relation between rotation,Vaand∆P forVa≈34 m /h 181 Figure B.2 – Relation between rotation,V˙aand∆P forV˙a≈51 m3/h 182 Figure B.3 – Working logic of the air flow rate control system . . . 184 Figure D.1 – Drawing and dimensions of the heat exchanger used in the
experiments . . . 195 Figure E.1 – Overlap of the frost layers on the radial fin and on the tube 197 Figure E.2 – Volume of frost added to each peripheral fin . . . 198 Figure E.3 – Volume to be subtracted due to the overlap of the frost
Table 4.1 – Characteristics of the PFT heat exchanger prototype used in the experiments . . . 90 Table 4.2 – Test conditions (nominal values). . . 92 Table 4.3 – Actual conditions of the experiments. . . 93 Table 6.1 – Comparison between simulation and experimental results
of the heat transfer rate, pressure drop and enthalpy effec-tiveness at the beginning and at the end of all tests . . . . 157 Table B.1 – Coefficients of equation (B.1) . . . 183 Table B.2 – Coefficients of equation (B.2) . . . 183 Table C.1 – Standard uncertainty of the variables directly measured . . 186 Table C.2 – Combined uncertainty of the temperature measurement
sys-tem . . . 186 Table C.3 – Determination of the combined uncertainty of the dry air
density . . . 189 Table C.4 – Determination of the combined uncertainty of the dry air
specific heat capacity at constant pressure . . . 190 Table C.5 – Determination of the combined uncertainty of the air
hu-midity ratio . . . 190 Table C.6 – Determination of the combined uncertainty of the air flow
rate . . . 191 Table C.7 – Determination of the combined uncertainty of the frost flow
rate . . . 192 Table C.8 – Determination of the combined uncertainty of the mass of
frost accumulated . . . 193 Table C.9 – Determination of the combined uncertainty of the sensible
heat transfer rate . . . 194 Table C.10–Determination of the combined uncertainty of the latent heat
transfer rate . . . 194 Table C.11–Determination of the combined uncertainty of the total heat
1 I . . . . 29 1.1 Motivation . . . 32 1.2 Objectives . . . 40 1.3 Document structure . . . 41 2 L . . . . 43
2.1 Enhancement of air-side heat transfer using peripheral finned-tube heat exchangers . . . 44 2.2 Measurement and correlation of frost physical properties . . 46 2.3 Performance of heat exchangers under frosting conditions . . 56 2.4 Closing remarks . . . 62 3 F . . . . 65 3.1 Heat transfer in wet surfaces . . . 65 3.2 Heat transfer in wet fins . . . 67 3.3 Overall heat transfer coefficient for a wet surface . . . 70 3.4 Pressure drop in porous media . . . 73 3.5 Heat transfer in porous media . . . 76
4 E . . . . 79
4.1 Experimental facility . . . 79 4.2 Test section . . . 81 4.3 Air flow rate control system . . . 83 4.4 Air conditioning system . . . 84 4.5 Fluid cooling system . . . 85 4.6 Experimental procedure . . . 86 4.7 Heat exchanger prototype . . . 87 4.8 Data regression . . . 91 4.9 Experimental conditions and preliminary test results . . . 92
5 M . . . . 97
5.1 Thermal model . . . 97
5.1.1 F . . . 97
5.1.2 H . . . 103
5.4 The influence of time on the heat exchanger parameters . . . 115 5.5 Computational implementation . . . 116
6 R . . . 121
6.1 Experimental results . . . 121 6.2 Mathematical model predictions . . . 126
6.2.1 F . . . 126
6.2.2 A - . . . 140
6.2.3 T . . . 146
7 C . . . 159 7.1 Recommendations for future work . . . 162
B . . . 165 A A G . . . 173 A B A . . . 181 A C E . . . 185 A D D . . . . 195 A E F . . . 197
Symbol Description Unit
A Area m2
cp Specific heat capacity at constant pressure J/kg◦C
d Diameter m
e Thermal conductance per unit volume W/m3◦C
f Friction factor
-h Enthalpy J/kg
ℏ Heat transfer coefficient W/m2◦C
ℏm Mass transfer coefficient kg/m2s
k Thermal conductivity W/m◦C K Permeability m2 Kc Contraction coefficient -Ke Expansion coefficient -L Length m ˙
m Mass flow rate kg/s
M Mass kg
N Number
-p Perimeter m
P Pressure Pa
q Heat transfer rate per unit length W/m
˙
Q Heat transfer rate W
˙ Q′′ Heat flux W/m2 R Thermal resistance ◦C/W t Time s t Thickness m T Temperature ◦C u Velocity m/s
U Overall heat transfer coefficient kg/m2s
V Volume m3
˙
V Volumetric flow rate m3/s
w Fin height m
˙
W Pumping power W
z Pumping power per unit volume W/m3
Greek symbols
Symbol Description Unit
β Heat exchanger surface area per unit volume 1/m
δ Frost thickness m ϵh Enthalpy effectiveness -ε Porosity -η Fin/surface efficiency -µ Viscosity kg/ms ρ Density kg/m3
σ Ratio of the minimum free-flow area and the face area of the heat exchanger
-τ Tortuosity
-ϕ Relative humidity
-ω Humidity ratio gH2O/kgdry air
Subscripts
Symbol Description a Air add Added b Bare tube B Base c Coolantcold Cold surface
comp Compressor
cond Condenser
cs Cross-section
D Darcy disc Discounted db Dry-bulb evap Evaporator exp Experimental f Fictitious F Fin
face Face area
fan Fan g ice h Hydraulic hx Heat exchanger i Internal in Inlet K Kozeny L Length LM Log-mean loss Loss max Maximum min Minimum mod Model n Nozzle o Overall, outer oc Effective out Outlet p Peripheral, particle P Pore r Radial rise Rise s Frost sat Saturation sd Solid
sm Solid matrix sub Subtracted sur Surface t Total
Abbreviations
Symbol Description CFC ChlorofluorocarbonCFD Computational Fluid Dynamics COP Coefficient of Performance
CV Control volume
EES Engineering Equation Solver GWP Global Warming Potential HCFC Hydrochlorofluorocarbon
HFC Hydrofluorocarbon
Le Lewis number
NTU Number of transfer units
Nu Nusselt number
PID Proportional-Integrative-Derivative PFT Peripheral finned-tube
Pr Prandtl number
1 I
Its been a long time since the mankind realized the benefits of keeping a specific location at a lower temperature than the surrounding environment, using it for food and medicine preservation and thermal comfort. According to Gosney (1982), “The science and art of refrigeration is concerned with the cooling of bodies or fluids to temperatures lower than those available in the surroundings at a particular time and place”. Many ways of producing cold have been invented and developed throughout history. The first one is related to harvesting snow and ice blocks (Figures 1.1a and 1.1b), used to keep food at low temperatures. In 1830, in the United States, Frederic Tudor, “The Ice King”, built his fortune selling ice for cooling purposes. He transported natu-ral ice extracted from New England to warmer locations, like the Caribbean islands, the southern states, Europe and even India (Anderson, 1953).
(a) United Stated Food Administration advertisement on ice harvesting
(b) Extraction of natural ice
Figure 1.1 – Harvesting of ice was a strong economic activity in the 1800’s The first mechanical refrigeration systems were developed using fluids like sulfur dioxide, ether, carbon dioxide, wine and vinegar. These machines entered the market as competitors to the ice extractors. The latter claimed that mechanical systems had constant failures, while the natural ice was always available (Briley, 2004). In 1834, Jacob Perkins patented and presented one of the first vapor compression refrigeration systems (Figure 1.2a) based on
an invention by Oliver Evans, who conceived a system that condensed and evaporated a fluid for continuously producing cold (Gosney, 1982). Later, in 1850, Edmond Carré developed the first absorption refrigeration system, using a brine that contained water and sulfur acid. In 1859, his brother, Fer-dinand Carré, patented and built a ice-making machine based on this system (Figure 1.2b) using water and ammonia in the absorption cycle.
(a) Perkin’s vapor compression system (b) Carré’s ice machine, which used an absorption refrigeration system
Figure 1.2 – First refrigeration systems
Today, different types of refrigeration technologies are used worldwide, such as mechanical vapor compression, gas cycle, thermo-electric, absorption, adsorption and, more recently, magnetocaloric. The majority of commercial systems is based on the mechanical vapor compression of synthetic (e.g., R-134a, R-410A) or natural (e.g., R-600a, R-290, R-744) refrigerants.
Figure 1.3 illustrates the standard vapor compression refrigeration cycle, and Figure 1.4 presents its pressure-enthalpy diagram. The cycle starts at point 1, where the fluid enters the compressor, having its pressure and temperature raised until it reaches the state indicated by point 2. Then, the fluid goes to the condenser, where it releases heat to a heat sink, having its temperature de-creased before changing phase from vapor to liquid, reaching state 3. The next step consists in throttling the sub-cooled liquid through an expansion device, where the pressure is reduced and the enthalpy is kept approximately constant, until it reaches state 4. Finally, the fluid goes to the evaporator, where it draws heat from the ambient or substance to be cooled (usually air), completing the
cycle. The primary performance index of a refrigeration system is the
coeffi-cient of performance (COP), defined as the ratio of the heat transfer rate in the
evaporator (cooling capacity) and the power consumption of the compressor:
COP = ˙ Qevap ˙ Wcomp (1.1) Expansion device Compressor Evaporator Condenser 2 3 4 1 ሶ 𝑄𝑐𝑜𝑛𝑑 ሶ 𝑄𝑒𝑣𝑎𝑝 ሶ 𝑊𝑐𝑜𝑚𝑝
Pressure Enthalpy ሶ 𝑄𝑐𝑜𝑛𝑑 ሶ 𝑄𝑒𝑣𝑎𝑝 ሶ 𝑊𝑐𝑜𝑚𝑝
3
2
1 4𝑃
𝑐𝑜𝑛𝑑 𝑃𝑒𝑣𝑎𝑝Figure 1.4 – P-h diagram of the standard vapor compression refrigeration cycle
1.1
M
According to estimates of the International Institute of Refrigeration, there are roughly 3 billion refrigeration, heat-pump and air conditioning systems operating worldwide, with a global annual sales of approximately 300 bil-lion US dollars (IIR, 2015). The refrigeration sector employs more than 12 million people and consumes about 17 % of the overall electricity produced worldwide. For instance, air-conditioning equipment are responsible for about 14 % of total electricity consumed in the United States, while 40 % in the city of Mumbai, India.
The concerns of society regarding energetic and environmental issues are growing continuously, with a clear consensus that environmental sustainabil-ity is a major goal to be achieved globally. Today, the most discussed envi-ronmental issue is Global Warming. This phenomenon is an escalation of the Greenhouse Effect, which is intensified by the large quantities of carbon diox-ide and other greenhouse gases dumped in the atmosphere, as a by-product of energy production, like burning mineral coal in power plants or gasoline and diesel in automobiles.
capaci-ties to absorb infrared radiation. In order to evaluate the capacity of each type of gas, a scale called Global Warming Potential (GWP) was created. It is a measure of the amount of heat trapped per unit mass of a given substance, com-pared to the emission of one ton of carbon dioxide, in a certain time period. One of the environmental footprints of cooling systems is the disposal and leakages (direct emission) of the high-GWP fluids (CFCs, HCFCs and HFCs) used in the refrigeration units. For example, the 100-year GWP of R-134a is 1300, which means that in a period of one hundred years, one ton of R-134a will absorb the same amount of radiation trapped by 1300 tons of CO2. There
has been a big effort by governments and corporations around the world to reduce the usage of these products. Usually, investments are made in research and development of technologies that use less harmful greenhouse gases, like CO2itself. Also, international treaties, like the Kyoto and the Montreal
Proto-cols, have served to promote and regulate new environmental policies related to the release of greenhouse gases in the atmosphere.
Using environmentally benign fluids is not the only way the refrigeration industry can help to slow down Global Warming, as about 80 % of the global-warming impact of refrigeration systems come from indirect emissions, orig-inated from the electricity production by fossil fuel power plants required to power the systems (Prata and Barbosa, 2009). Developing better technologies to increase the efficiency of cooling systems has a significant impact on the energy consumed by these devices and, on a big scale, indirectly leads to a reduction of the greenhouse gases disposed in the atmosphere.
Although the compressor is, by far, the most energy intensive component of a refrigeration system, improving the design of other components, such as the heat exchangers, may lead to the reduction of material and operating costs associated with the system as a whole. For example, it is well known that frost formation in evaporators (Figure 1.5) depletes the thermal and hy-draulic performance of direct expansion refrigerators and heat pumps. When the temperature of the evaporator surface is lower than the freezing tempera-ture of water, the water vapor in the surrounding air desublimates on the cold heat exchanger surface. This process generates an additional low-conductivity thermal resistance on the coil, which increases continuously as the frost grows
thicker. In fan-supplied evaporators, another negative effect of frosting is the decrease of the air flow rate caused by the blockage of the channels. Both phe-nomena increase the energy consumption of the cooling system by forcing the fan and the compressor to run for longer periods.
Figure 1.5 – Frost formation over an evaporator surface
There has been a substantial effort by researchers and institutions world-wide to improve the performance of cooling systems. Searching for more effi-cient heat transfer technologies is essential to reduce the energy consumption on these devices. Over the past few decades, the importance of heat exchanger design has increased immensely from the viewpoint of energy conservation, conversion, recovery, and successful implementation of new energy sources. Its importance is also increasing from the standpoint of environmental con-cerns such as thermal pollution, air pollution, water pollution and waste dis-posal (Shah and Sekulic, 2003).
The evaporator has a noteworthy position in this context, since it is respon-sible for providing the desired cooling capacity and lowering the temperature to the specified design level. In direct expansion systems, the largest thermal resistance is usually on the air side (natural or forced convection). Therefore, to increase the overall thermal conductance, one solution has been to increase the external (air-side) surface area by means of extended surfaces (fins). The heat exchanger geometry and the amount of extra material on the air side is dictated by design constraints, such as volume, weight or cost.
In refrigeration and air conditioning, the most common tube shapes used in heat exchangers are the round tube and the flat (round-edge rectangular) tube. As for the fins, tube-fin exchangers can be classified as (1) individu-ally finned or (2) flat (continuous) fins (Figure 1.6). As presented in Figure 1.7, the morphology of the fins can be classified as (a) plain, (b) wavy, (c) louvered, among others. Louvered fins are designed to interrupt the air flow boundary layer, generating vortices to intensify the heat transfer. The pres-ence of fins, however, has the inconvenipres-ence of adding a resistance to the air flow, increasing the air-side pressure drop. Also, it is difficult to remove foul-ing accumulated on the fins, such as dust particles. Frost can be considered as an intermittent type of fouling, whose rates of accretion and removal are related to the heat transfer rate itself.
Flow Flow
(1) (2)
Figure 1.6 – (1) Individually finned tubes; (2) flat (continuous) fins on an array of tubes (Shah and Sekulic, 2003)
Increasing the surface area density in a heat exchanger reduces the com-ponent size for a fixed thermal duty. In a recent study, Wu et al. (2007) in-troduced a new type of air-side enhanced surface, the so-called peripheral finned-tube (PFT). The peripheral finned-tube geometry consists of six radial
(a) (b) (c)
Figure 1.7 – (a) plain fins; (b) wavy fins; (c) louvered fins (Mao et al., 2013) fins evenly spaced around the tube, which are connected to each other at their tips by peripheral fins, creating a hexagon shaped set formed by six radial fins and six peripheral fins. The sets are manufactured in three different sizes (also known as “levels”) and mounted around the tubes with an angle offset of 30° between each other, as illustrated in Figure 1.8. The hexagonal fin sets form a bed of interconnected pores, which allows the air to flow normal to the tubes, but through alternative paths in case there is a blockage in a specific part of the heat exchanger.
The PFT geometry is intended to be more robust against condensate or frost formation than conventional tube-fin heat exchangers (Wu et al., 2007). The regions around the tubes are the coldest part of the heat exchanger, so frost would have a preference to grow there, as illustrated in Figure 1.9 from Wu
et al. (2007). Since the regions around the tubes are flow stagnation zones,
the flow would be less disturbed by the frost growth, and would be more easily redistributed through alternative paths through the porous matrix, as shown in Figure 1.10. According to Wu et al. (2007), the anisotropy of the porous structure also facilitates condensate drainage. It should be noted that the frost blockage shown in Figure 1.9 is a subjective representation of the frost formation process, and has not been obtained from the application of a mathematical model.
Figure 1.11 compares the thermal-hydraulic performance of the peripheral finned-tube geometry with conventional geometries, namely plain, wavy and louvered, in terms of the thermal conductance per unit volume as a function of the required pumping power per unit volume. In addition, the “no frost” geom-etry (Barbosa et al., 2009) was included in the comparison. This geomgeom-etry is
Figure 1.8 – Peripheral finned-tube geometry blockage blockage blockage air inlet air outlet
Figure 1.9 – Representation of the blockage caused by frost in the PFT geometry. Adapted from Wu et al. (2007)
commonly used as evaporators in household refrigerators and light commer-cial appliances, and consists of an aluminum tube-fin heat exchanger with plain aluminum fins with variable spacing between them. The larger spacing between the fins, typical of “no frost” evaporators, reduces the surface area per unit volume, but is necessary to accommodate large quantities of frost on the air side.
Figure 1.10 – Air flow across the peripheral fins (Wu et al., 2007) The thermal conductance per unit volume is given by (Shah and Sekulic, 2003)
e = ηoℏoβ (1.2)
whereηo is the overall surface efficiency,ℏois the air-side heat transfer co-efficient andβis the surface area per unit volume. The pumping power per unit volume can be written as (Shah and Sekulic, 2003):
z = 2fouD 3ρ
a dh
(1.3) wherefo is the air-side Fanning friction factor,uD is the air face (Darcian) velocity,ρais the air density anddhis the hydraulic diameter given by
dh = 4Amin
pf ace = 4σ
β (1.4)
whereAminis the minimum free-flow area,pf aceis the perimeter of the face area of the heat exchanger andσ is the ratio of the minimum free-flow area and the face area of the heat exchanger. As will be seen, in the PFT heat exchanger,σ coincides with the porosity of the matrix. In Figure 1.11, the
z [W/m3] 100 101 102 103 104 e [W/m 3 ° C] 103 104 105 106
PFT - Whitaker et al., 1972; Ergun, 1952 Louvered - Wang et al., 1999
Wavy - Wang et al., 1997 Plain - Wang et al., 1996 No frost - Barbosa et al., 2009
Figure 1.11 – Thermal conductance per unit volume as a function of the pumping power per unit volume for five types of air-side
enhanced surfaces
thermal conductance and the pumping power were evaluated using state-of-the-art correlations for heat transfer and pressure drop in enhanced surfaces obtained from the literature and indicated in the figure legend. The correla-tions used to predict the performance of the peripheral finned-tube geometry have been validated in the experimental work of Pussoli (2010).
For consistency, the overall surface efficiency,ηo, has been assumed equal to unity in the present calculations. Also, the geometrical parameters (e.g., fin dimensions and pitch) required in each correlation have been taken as the values which provided the largest transfer coefficients within the ranges of validity of each parameter pertaining to each correlation. This guaranteed the
best possible performance of each air-side geometry.
As can be seen from Figure 1.11, the louver fins have a higher thermal conductance per unit volume for the same pumping power per unit volume than the other conventional fin types. This was somewhat expected, as was the worst performance of the “no frost” geometry because of its much lower area density.
The peripheral finned-tube geometry presented the highest thermal con-ductance for the same pumping power per unit volume. However, for the same range of face velocities used to produce thee-zcurves, the specific pumping power of the new air-side geometry was one order of magnitude (or more) higher than those of the conventional geometries. This places the peripheral finned-tube geometry as a competitive alternative to conventional enhanced surfaces in applications where there is a need for high rates of heat transfer per unit volume without a severe limitation on the air-side pumping power.
1.2
O
The objective of this work is to investigate the thermal-hydraulic perfor-mance of peripheral finned-tube heat exchangers under frosting conditions. The present analysis complements previous work carried out by Pussoli (2010) on the evaluation of the thermal-hydraulic performance of peripheral finned-tube heat exchangers under dry conditions, i.e., without frost or condensate formation on the air side. To accomplish this objective, the research was di-vided in two parts. In the experimental part, one of the peripheral finned-tube heat exchanger prototypes developed by Pussoli (2010) was tested in a closed-loop wind-tunnel calorimeter specially designed to measure the performance of heat exchangers under frosting conditions (Silva, 2012). Different combi-nations of operating parameters, such as coolant temperature, inlet air temper-ature, inlet air relative humidity and air flow rate have been evaluated. The modeling part consisted of developing a mathematical model of the heat and mass transfer processes in the peripheral finned-tube geometry, and using this model to predict the accumulated frost mass, pressure drop and heat transfer behavior of the heat exchanger as a function of time.
1.3
D
This dissertation was structured in seven chapters. After the present in-troductory chapter, the second chapter reviews the state of the art on frost formation in heat exchangers. The third chapter presents the theoretical fun-damentals of heat and mass transfer on wet surfaces, which are the basis of the mathematical model presented in chapter 5. The fourth chapter is focused on the experimental work, and describes the experimental facility, the heat ex-changer prototype and the experimental procedure. The fifth chapter presents the development of the mathematical model for predicting frost formation in the peripheral finned-tube heat exchanger. In chapter 6, the experimental and modeling results are presented and discussed. In chapter 7, final conclusions are drawn and recommendations for future work are made.
2 L
Over the past few decades, a lot of effort has been put into understand-ing the frost formation process. Although ice and frost contain water in its solid state, the two media are quite different, but commonly confused. As illustrated in Figure 2.1, ice is the result of the phase change process repre-sented in line 1, through which the liquid solidifies. In a heat exchanger, ice is formed when the surface temperature is below the freezing point of water, but the dew-point of the air is above the freezing temperature of water. On the other hand, frost is the result of process represented in line 2, which consists of the desublimation of the water vapor into solid form. These two processes re-sult in different properties of the rere-sulting media, most important, from a heat transfer point of view, the density, specific heat capacity and thermal conduc-tivity. Ice has a massive and denser structure, different from frost, which is formed by irregular-shaped ice crystals and air, resulting in a non-structured porous medium (Iragorry et al., 2004).
Fusion curve Sublimation curve Solid Liquid Vapor Triple point Critical point Temperature [K] Pre ssu re [kPa ] 22064 647.01 273.16 0.612 1 2
This chapter is divided into three main parts, each related to one aspect of the research carried out in this dissertation. Section 2.1 deals with the devel-opment of the peripheral finned-tube geometry. Section 2.2 presents studies on measurement of frost properties and their correlation, in which the main objective has been to develop methods to compute frost growth and densifica-tion. Section 2.3 reviews several works on frost formation in heat exchangers, and how it affects their performance.
2.1
E
-The peripheral finned-tube geometry is a new type of air-side surface enhancement. Detailed descriptions of conventional fin geometries used in liquid-gas heat transfer enhancement (Figures 1.6 and 1.7) have been given by Shah and Sekulic (2003) and Webb and Kim (2005) and will not be re-peated here for brevity.
Wu et al. (2007) carried out the first study of the peripheral finned-tube geometry. In addition to developing an analytical heat conduction model to calculate the temperature distribution in the radial and peripheral fins, the authors carried out a Computational Fluid Dynamics (CFD) investigation of the air flow through the porous matrix. The simulations revealed the occur-rence of higher heat transfer rates through the peripheral fins when there is no frost formation (dry conditions). Also, the heat transfer rate was lower in the regions closer to the tube, where there is less air motion. The effect of frost formation was simulated by adding a material to the region around the tubes with values of thermal conductivity and density equal to those of frost. The results were very close to the case without any frost, with a reduction of approximately 1.7 % of the heat transfer rate and an increase of 3 % of the pressure drop, indicating that the blockage caused by the frost had little influence, since the air is stagnant near the tubes. In both cases, the heat trans-fer was more intense through the peripheral fins, and the radial fins serve as a path for the heat to flow to the refrigerant, keeping a close match of the performance of the evaporator with or without frost formation.
Pussoli et al. (2012a) performed the first work involving an experimental analysis of peripheral-finned tube heat exchangers, aiming to determine the heat transfer and pressure drop characteristics on the air side. In their work (also presented in Pussoli, 2010), five heat exchanger prototypes were devel-oped with different geometries, namely different values of radial fin length, fin distribution, number of fins and number of tube rows. The heat exchangers were tested in an open-loop wind tunnel, with air flow rates ranging from 30 to 110 m3/h. A mathematical model of the heat exchanger was developed, which consisted of two sub-models. The first sub-model quantified the heat transfer through the fins, by calculating the temperature profile in the radial and pe-ripheral fins. The second sub-model calculated the energy and momentum balances in one-dimensional control volumes to determine the pressure drop and air temperature variation along the heat exchanger. The temperature dis-tributions calculated in the first sub-model were used to determine the overall surface efficiency in the second. Correlations for the interstitial Nusselt num-ber and friction factor in porous media were used to provide closure to the one-dimensional model. Very similar overall agreement between the model and the experimental data was obtained using the correlations of Whitaker (1972) and Handley and Heggs (1968) for the particle-diameter based Nus-selt number and of Ergun (1952) and Montillet et al. (2007) for the friction factor. The experimental analysis showed that the lowest values of thermal conductance were found in the heat exchangers with the largest face areas, which resulted in low air velocity through the matrix and, consequently, low heat transfer coefficients. In most cases, the overall surface efficiency was above 95 %. The mathematical model results for the heat transfer rate and air-side pressure drop were in good agreement with the experiments, remaining within the uncertainty of the experimental data for the former. For the air-side pressure drop, the model underestimated the experimental data at some conditions, reaching deviations of up to 30 %.
In a subsequent study, Pussoli et al. (2012b) combined the mathematical model for the air-side heat transfer pressure drop with the entropy generation minimization method of Bejan (1996) to determine the optimal characteris-tics of peripheral-finned tube heat exchangers. The analysis was based on the
three performance evaluation criteria (PEC) for single-phase convection pro-posed by Webb and Kim (2005), namely fixed geometry, fixed face area and variable geometry. Limiting operating (boundary) conditions of constant heat flux and constant wall temperature were investigated. Given the large number of geometrical parameters associated with the peripheral finned-tube heat ex-changers, the work considered the solid matrix porosity, the equivalent parti-cle diameter and the partiparti-cle-based Reynolds number as the main independent parameters dictating the fluid flow and the heat transfer through the porous medium. The analyses were performed for a fixed heat transfer rate of 300 W, and resulted in combinations of the independent parameters which yielded a minimum entropy generation rate according to the operating condition.
An important conclusion from the works of Pussoli and co-workers was that the PFT geometry provides a large area density (surface area per unit volume), which is good in terms of thermal performance. However, this high surface density also results in a large pressure drop, which may limit the ap-plicability of the peripheral finned-tube geometry due to the high pumping power required to promote the air flow.
2.2
M
-Understanding frost formation and growth is important in many engineer-ing applications. Frostengineer-ing is usually undesired, for it is generally linked to a performance impairment. In aircrafts, anti-icing systems are needed to prevent frost and ice accretion on the external surfaces, which would otherwise change the aerodynamic forces on the wings and lessen the measuring capabilities of sensing equipment (Souza et al., 2016). In refrigerators and heat pumps, frost-ing in the evaporator increases the overall thermal resistance, which decreases the cooling/heating capacity and reduces the COP. In household refrigerators, condensation and frosting on the inner liner of the refrigerated compartments is an important design concern motivated by the consumer perception of the product (Piucco, 2008).
and mass transfer processes within a non-structured porous medium. Also, the resulting solid can be formed in a variety of internal structures and shapes. As presented in Figure 2.2, different types of ice crystal shapes can develop, de-pending on the supersaturation degree (the humidity ratio difference between the air stream and the cold surface) and the temperature of the cold surface (Kobayashi, 1958). This, in turn, affects frost thermophysical properties, such as the thermal conductivity (Na and Webb, 2004a). Several authors have inves-tigated the processes of frost formation and growth, in an attempt to correlate them with operating parameters, such as air humidity and temperature, flow velocity and surface conditions (temperature and finishing).
Figure 2.2 – Ice crystal shapes as a function of local conditions (Libbrecht, 2005)
The works on frost formation prediction can be divided in two categories, depending on the physical assumptions. Some authors claim that the assump-tion that air is saturated at the frost-air interface (saturaassump-tion model) is
funda-mentally wrong. However, this assumption is still adopted in some cases for simplicity (Lee et al., 1997; Lee et al., 2003; Yang and Lee, 2005; Kandula, 2011; Kim and Lee, 2014; Hwang and Cho, 2014). Na and Webb (2004a) have shown that the water vapor is supersaturated at the frost surface. More recently, supersaturation models have been developed by other authors (Na and Webb, 2004b; Hermes, 2012; Cheikh and Jacobi, 2014; Loyola et al., 2014).
Hayashi et al. (1977) performed an experimental investigation of the frost formation on a flat plate, aiming to identify the relation between the proper-ties of the flow and of the porous medium. The frost growth was observed to depend on the air humidity and temperature, cold surface temperature and air flow velocity. The process was divided in three phases: (i) crystal growth, (ii) frost layer growth and (iii) complete frost layer growth. In the first phase, the cold surface is covered by a thin frost layer, where the crystals grow vertically and distant from each other. The second phase consists of the ramification of the ice crystals due to water vapor diffusion. The frost layer is filled by the va-por, and its thickness increases continuously, until it reaches the next phase, which starts when the surface temperature of the frost reaches 0◦C. Then, instead of sublimating, the air vapor starts to condense on the surface. The liquid infiltrates the porous frost and freezes when it reaches a temperature lower than the freezing point of water. This process increases the density and the thermal conductivity of the porous medium, increasing the frost forma-tion rate. The authors suggest that the frost density increases parabolically with time, due to the gradual reduction of the thickness growth rate (which increases with the square-root of time). As a result, the mass deposition rate remains almost constant. The expression developed for the frost density is given by
ρs= 650 exp(0.277 Ts) (2.1) The effective thermal conductivity was estimated based on a model that con-sidered the frost formed by solid columns of ice and air, and good agreement was found with experimental data.
Gall et al. (1997) developed a one-dimensional transient model for the conjugate heat and mass transfer problem in a frost layer. The model was based on the local volume averaging technique, which considers the humid air and ice to coexist in each elementary control volume, and calculates the time-dependent temperature and density distributions in the frost layer. They observed that several available correlations for the water vapor diffusivity did not represent the mass transport problem satisfactorily, since the values obtained in their experiments were much higher than the predictions of the classic diffusion theory. The authors then proposed a new correlation for the water vapor diffusion resistance factor.
Lee et al. (1997) developed an analytical model for predicting frost growth on a cold flat surface, considering the molecular diffusion of water and heat generation due to the sublimation of water vapor in the frost layer. An exper-imental apparatus (a closed-loop wind tunnel) was used to perform tests to validate the theoretical analysis. Comparisons of the model with the experi-mental results presented an average error of 10 % for the frost thickness. The authors also noticed that the frost thickness and surface temperature increased with increasing air velocity and relative humidity.
Cheng and Cheng (2001) developed a theoretical model to simulate the frost formation process on a cold flat plate by modifying earlier models pro-posed by Jones and Parker (1975) and Sherif et al. (1993). Cheng and Cheng (2001) took into account the influences of the cold surface temperature, air velocity, air humidity and air temperature. They noticed that the frost growth is relatively faster in the early stages of the simulations and slows down as times passes. This relatively high growth rate was attributed to a lower ther-mal resistance associated with a thinner frost layer. Besides, as the density of the frost layer is lower in the early stages, the deposition of a small amount of water vapor causes a large increase in frost thickness. The predictions of the model agreed more closely to the experimental data provided by Yonko and Sepsy (1967) than the simulation results by Jones and Parker (1975) and Sherif et al. (1993). However, for some particular cases, the model by Sherif
et al. (1993) may give more accurate results.
litera-ture about frost properties, correlations and mathematical models. The authors divided the frosting research into two general groups: experimental correla-tions and mathematical models. In general, the properties correlated are the frost thermal conductivity, average density and air-frost heat transfer coeffi-cient, and these parameters are often determined for a limited operation range. The mathematical models include both differential and integral approaches, and are classified based on the geometry of the external surface. According to the authors, it is clear that the recent studies focus on distributed analyses in which the frost properties and parameters vary with time and within the frost thickness. As for the process modeling, most of the differential models consider a constant frost surface density as a boundary condition, which may lead to an under estimation of frost porosity and vapor mass diffusion, and, therefore, over estimation of the frost layer thickness.
Na and Webb (2004b) developed a mathematical model to predict frost de-position and growth. The authors stated that all previous works assumed the water vapor to be saturated at the frost-air interface, when the fundamentally correct condition is supersaturated water vapor at the interface. The model was based on improved correlations for frost thermal conductivity developed by the authors, a rationally based tortuosity factor and was able to calculate the local frost density variation. The frost growth rate was measured experi-mentally and compared with the model results, resulting in an agreement of 15 %. If the air was to be considered saturated, the model would over predict the frost growth rate by approximately 30 %. Experimental data from three in-dependent sources were compared with the model, and the authors concluded that the supersaturation model is superior to the models which consider the water vapor as saturated at the air-frost interface.
Hermes et al. (2009) developed a theoretical and experimental work to investigate frost growth and densification on plain surfaces. A small-scale open-loop wind tunnel was built, in which the air temperature and humidity were controlled at the test section inlet to simulate the typical operating condi-tions of domestic refrigerators. A cold aluminum plate in the test section was responsible for creating the conditions for the frost build up. An image acqui-sition system was installed to photograph the frost formation, thus allowing
the evaluation of the frost thickness evolution with time. The authors used the experimental data to derive an empirical correlation for the frost density, introducing a third coefficient in the correlation of Hayashi et al. (1977) to account for the plate temperature:
ρs= 207.3exp(0.266 Ts− 0.0615 Tcold) (2.2) where Tsis the frost surface temperature andTcold is the cold plate surface temperature. The new frost density correlation agreed to within±15 % with the experiments. The correlation was combined with mathematical models developed by other authors (Lee et al., 1997; Cheng and Cheng, 2001; Na and Webb, 2004b) to develop a model for frost growth. The model presented good agreement with the experimental data, as the calculated frost thickness was within the experimental uncertainty.
Kandula (2011) developed a mathematical model to investigate the mean parameters that affect the frost formation on flat surfaces. He correlated the frost density and thermal conductivity over wide ranges of frost formation conditions. Among the simplifications adopted, the mass and energy transfer processes were considered to be one dimensional, transverse to the cold wall, and the air condition was considered saturated at the frost-air interface. The model was compared with experimental data available in the open literature (Yonko and Sepsy, 1967; Lee et al., 1997; Cheng and Wu, 2003; Lee et al., 2003; Hermes et al., 2009), with a satisfactory agreement.
Hermes (2012) developed an analytical model based on macroscopic heat and mass balances to predict frost growth and densification on flat surfaces. The model was based on a dimensionless formulation that resulted in an al-gebraic expression for the frost growth as a function of the Nusselt number of the air flow, the supersaturation degree and the temperature difference be-tween the air and the cold surface. The model predictions of the frost layer thickness were compared to experimental data obtained from works available in the open literature and a good agreement was observed. The author claimed that the method developed has the advantage of clearly indicating the main pa-rameters that dictate frost formation, is easy to implement and requires very low computational effort, being suitable to be used for predictions of frost
formation on complex geometries, such as tube-fin heat exchangers or even aircraft wings. However, the model validity still needs to be verified for a wider range of conditions.
Cheikh and Jacobi (2014) presented a mathematical model to calculate the growth and densification of a frost layer on flat surfaces, focusing on impos-ing physically realistic boundary conditions. Accordimpos-ing to the authors, most mathematical models of frost formation rely on solving the heat and mass dif-fusion equations in the frost layer. They can be classified in two categories, depending on the chosen boundary conditions: (i) saturation models, in which the water vapor is assumed to be saturated at the cold surface and at the frost-air interface, and (ii) supersaturation models, in which the water vapor at the frost-air interface is assumed to be supersaturated. To calculate the frost perature, convection at the air-frost interface and prescribed cold plate tem-perature conditions were used. To calculate the water vapor density at the interface, a constant heat flux condition was used at the cold surface. The au-thors claim that considering the water vapor as saturated does not have any physical basis and that these hypotheses were not validated experimentally, and for that reason is was their intention to check the validity of the satura-tion boundary condisatura-tions. The saturasatura-tion model results for frost density and thickness were within the expected range. However, the water vapor density at any location within the frost layer was lower than the saturated water-vapor density at the corresponding local temperature. In this situation, mass transfer would occur from the frost layer to the surrounding air. As the frost layer is growing, this situation is impossible. In order to validate the model, a series of experiments under different conditions were performed in a wind tunnel. The results by the model for the frost density and thickness were within±10 % of the experimental data.
Hermes et al. (2014) developed a mathematical model to predict the evo-lution of a frost layer porosity with time. The authors claim that even with an abundant literature related to frost formation, most models rely on empirical correlations for the frost density, which limits their applications. A theoretical model was developed for the frost porosity, which was adjusted using exper-imental data to derive a semi-empirical correlation for the frost density as a
function of time and of the modified Jakob number, a dimensionless parameter that relates the thermodynamic conditions of the air and cold surface. Differ-ent from most available models, the correlation was independDiffer-ent of the frost surface temperature, a variable that is usually hard to determine and requires sophisticated simulation models. Using experimental data of a previous work (Hermes et al., 2009), the validity of the model was considered satisfactory, as the results obtained were within±10 % error bounds.
Loyola et al. (2014) developed a mathematical model for frost formation in parallel-plate channels based on heat, mass and momentum balances con-sidering supersaturated air-frost interface conditions. The model results were compared with experimental data obtained by the authors for different air ve-locity, temperature, humidity and surface temperature conditions. Tests were conducted with durations of 30, 60, 90 and 120 minutes. It was observed that the supersaturation model predictions agreed satisfactorily with the experi-mental data, while the saturated air-frost interface condition had a tendency of over predicting the frost thickness.
Kim and Lee (2014) studied the applicability of the thermal network anal-ysis (TNA) technique in predicting frost growth on cold fins. Based on the frost formation model of Lee et al. (1997) and the Lewis analogy, the authors evaluated the time variation of the frost layer thickness and density. Also, the temperature profile of the frost layer, air stream and the fin were obtained for each time step. The results were compared with CFD simulations and exper-imental data for two and three dimensional cases reported by other authors. For one of the two-dimensional cases, the differences between the simula-tions and the experiments were 2.7 % for the surface temperature, 11 % for the frost thickness and 6.9 % for the frost density. The CFD analysis required approximately 93 minutes, whereas the TNA technique required only 9 sec-onds. Similar results were found for two three-dimensional cases, where the deviations of the frost thickness were 5.4 % and 12.2 %. Although the TNA technique presented a lower accuracy than the CFD analysis, the authors re-ported that this technique has the advantage of having lower computational costs for a quick estimation of the frost layer growth.
of frost growth and densification in parallel-plate channels. A scaling analysis of the problem was carried out to investigate the relationship between the frost growth and densification rates with the parameters involved. The study showed that both thickness and density obey the t1/2 scale, which leads to the conclusion that the accumulated frost mass varies linearly with time. A mathematical expression was developed to predict the frost density:
ρs= 2.2Λ(−3/4)t (1/2)
(2.3) where Λis the modified Jakob number, a dimensionless parameter that ex-presses the ratio between the supercooling and the supersaturation degrees and t is time. The supercooling degree is defined as the difference between the cold surface temperature and the dew-point. To confirm the predictions of the scaling analysis, experiments were performed in a closed-loop wind tun-nel. The air psychrometric conditions were controlled at the inlet of the test section, which was formed by two parallel aluminum surfaces to simulate the cold plates. Thirteen experiments were conducted varying the air temperature, air humidity, air velocity and plate temperature. A good agreement was found between the experimental data and the predicted frost density, with 85 % of the data remaining within±15 % error bands.
Kim et al. (2015) developed a model based on CFD to predict the macro-scopic and local frost behavior on a cold plate. A two-phase model was em-ployed to represent the air and the frost layer in each control volume in terms of their volume fractions. The experimental data of Lee et al. (2003) were used to validate the simulated average frost thickness, average frost density and frost surface temperature. The numerical results for the frost thickness agreed well with the experimental data, with maximum deviations of approxi-mately 0.3 mm during the early stages of frost formation (first thirty minutes). The model tended to over predict the layer thickness during the early stages. However, the results were considered satisfactory, considering the uncertainty of 6 % of the experiments. The same behavior was observed when calculat-ing the average frost density, as the model tended to over predict the data during the early stages of frost formation, with a better agreement when time approached 60 minutes. The influence of operating parameters on the frost
thickness, such as the air velocity, air humidity and cold plate temperature was also investigated. The influence of the cold plate temperature was vali-dated using the experimental results of Hermes et al. (2009), for which the temperature ranged from−16◦C to−4◦C. The experimental data of Lee et
al. (1997) were used to validate the model when the relative humidity varies
between 50 % and 80 % and air velocities from 0.5 to 2 m/s. A good agree-ment was also found in these cases, with maximum deviations of 10 %.
Negrelli and Hermes (2015) presented a semi-empirical correlation for the frost thermal conductivity. The authors used 188 experimental data points obtained from the open literature to fit an equation, valid for porosities ranging from 0.5 to 0.95. The development of the equation considered the dependence on the wall surface temperature by fitting the model to different temperature ranges (−30◦C to−4◦C), and the dependence on the supersaturation degree by means of the frost porosity. The correlation predictions were compared with experimental data with 81 % of the data points lying within the 15 % error band. Also, the correlation presented better prediction capabilities than the correlations proposed by several other authors.
Brèque and Nemer (2016) performed a study of the models available for frosting prediction. The authors claim that all the available models based on heat and mass diffusion through the frost layer use the same set of basic equa-tions. These include thermodynamic equilibrium at the frost surface (Gall et
al., 1997), frost layer with negligible thermal capacitance (Lee et al., 1997),
frosting modeled as a quasi-steady state phenomenon (Kondepudi and O’Neal, 1993), among others. However, many different assumptions are considered in each model, which can result in very different frost growth predictions. For instance, the analysis of the humidity ratio profile, effective thermal conduc-tivity and convection correlations have large impact on the results obtained for frost thickness. Thus, the assumptions have to be carefully chosen, depending on the case being studied. Some assumptions appear to be more physically realistic than others, and their incorrect selection can result in large errors (even triplicating the frost thickness predictions). With that, the authors ex-plained the effects of each assumption, and made recommendations to select the proper ones.
